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(2x^4+x^3-3x^2+5x-2) (x^2-x+1)

3 min read Jun 16, 2024
(2x^4+x^3-3x^2+5x-2) (x^2-x+1)

Multiplying Polynomials: (2x^4+x^3-3x^2+5x-2) (x^2-x+1)

This article will guide you through the process of multiplying the two polynomials: (2x^4+x^3-3x^2+5x-2) and (x^2-x+1).

Understanding the Process

Multiplying polynomials involves applying the distributive property multiple times. Each term in the first polynomial is multiplied by each term in the second polynomial.

Step-by-Step Solution

  1. Expand the First Polynomial: (2x^4+x^3-3x^2+5x-2)

  2. Expand the Second Polynomial: (x^2-x+1)

  3. Multiply Each Term in the First Polynomial by the Second Polynomial:

    • 2x^4 * (x^2-x+1) = 2x^6 - 2x^5 + 2x^4
    • x^3 * (x^2-x+1) = x^5 - x^4 + x^3
    • -3x^2 * (x^2-x+1) = -3x^4 + 3x^3 - 3x^2
    • 5x * (x^2-x+1) = 5x^3 - 5x^2 + 5x
    • -2 * (x^2-x+1) = -2x^2 + 2x - 2

  4. Combine Like Terms:

    • 2x^6
    • -2x^5 + x^5 = -x^5
    • 2x^4 - x^4 - 3x^4 = -2x^4
    • x^3 + 3x^3 + 5x^3 = 9x^3
    • -3x^2 - 5x^2 - 2x^2 = -10x^2
    • 5x + 2x = 7x
    • -2

  5. Write the Final Result:

(2x^4+x^3-3x^2+5x-2) (x^2-x+1) = 2x^6 - x^5 - 2x^4 + 9x^3 - 10x^2 + 7x - 2

Conclusion

By following these steps, we have successfully multiplied the two given polynomials, resulting in the polynomial: 2x^6 - x^5 - 2x^4 + 9x^3 - 10x^2 + 7x - 2. This process demonstrates the systematic approach to multiplying polynomials.

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